Problem 36 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ a^{-6} $$

Step-by-Step Solution

Answer

The simplest form of \(a^{-6}\) is \(\frac{1}{a^6}\).

Step 1: Understand Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. So instead of maintaining the negative exponent, we will turn the expression into a fraction to convert the exponent into a positive one.

Step 2: Rewrite the Expression

The given expression is \(a^{-6}\). To convert it to a positive exponent, rewrite it as the reciprocal: \(\frac{1}{a^{6}}\). This uses the property \(x^{-n} = \frac{1}{x^n}\).

Step 3: Simplify the Expression

The expression \(\frac{1}{a^{6}}\) is already in its simplest form, as we have converted the negative exponent to a positive by expressing it as a reciprocal.

Key Concepts

Reciprocal

When we talk about the concept of a **reciprocal**, we are referring to the inverse of a number or expression. In simpler terms, the reciprocal of a number is what you multiply it by to get 1. For instance, the reciprocal of 5 is \( \frac{1}{5} \), because \( 5 \times \frac{1}{5} = 1 \).

Applying this to variables and powers, the reciprocal plays a crucial role in dealing with negative exponents. When you encounter an expression with a negative exponent like \( a^{-6} \), think of it as a command to flip the base into the denominator and make the exponent positive, hence transforming \( a^{-6} \) into \( \frac{1}{a^6} \).

This is based on the rule that states: \( x^{-n} = \frac{1}{x^n} \). It's an essential property that helps us simplify expressions without sticking to negative powers. Remember, each time you convert a negative exponent to a positive one, you're essentially finding the reciprocal of the expression.

Positive Exponents

Turning negative exponents into **positive exponents** is about creating expressions that are simpler and easier to handle. A positive exponent indicates how many times we multiply the base by itself. For example, \( a^3 \) indicates multiplying \( a \) by itself three times: \( a \times a \times a \).

Negative exponents can sometimes seem confusing, but by realizing that they indicate a reciprocal action instead of just straightforward multiplication, you can transform them to positive exponents. For instance, the expression \( a^{-6} \) becomes \( \frac{1}{a^6} \), as mentioned before.

This conversion process is not just a mathematical exercise; it's a way to ensure that expressions can be manipulated with ease, whether you're adding, subtracting, multiplying, or dividing later on in calculations. Positive exponents are the norm for simplified and standard mathematical expressions.

Simplifying Expressions

In mathematics, **simplifying expressions** is key to making them more understandable and manageable. By reducing expressions to their simplest forms, calculations become straightforward and much less prone to error. The expression \( \frac{1}{a^6} \) is an example of a simplified expression using positive exponents.

Simplifying involves steps such as cancelling out terms, combining like terms, and, as in our case, transforming negative exponents into positive ones by utilizing reciprocal rules.

To achieve simplification, you should always look for opportunities to condense an expression without changing its value. For instance, turning \( a^{-6} \) into \( \frac{1}{a^6} \) renders the expression in a form that is easily recognizable and usable in further calculations. Remember, simplicity in mathematics is not just about ease; it's about clarity and precision as well.